Introduction

Summary

The main theme of the trimester was the interplay of different methods used in modern number theory. We wish to emphasize the new results and conjectures in arithmetic geometry, having direct bearing on the classical number theoretic problems. Two sessions, on the recently proved Serre's conjecture from 15 January to 14 February (organizers: L. Dieulefait and J.-P. Wintenberger) and on counting rational points on algebraic varieties from 15 March to 14 April (organizer: D.R. Heath-Brown), as well as a couple of shorter workshops, several seminars, and mini-courses were organized. The trimester culminated in a research conference from 15 to 19 April.

The aim of the session “Serre's conjecture” was to report on recent works linked to that conjecture, in particular about Galois representations and automorphic representations. During the weeks starting on 14 January and 21 January, Henri Carayol lectured on his work on the algebraic properties of Griffiths-Schmid varieties. The Griffiths-Schmid varieties are analytic varieties classifying Hodge structures. Studying their algebraic properties might be a step towards constructing Galois representations associated to automorphic representations appearing in the cohomology of these varieties. Our secondtheme related to the recent work of Michael Harris, Kai-Wen Lan, Richard Taylor and Jack Thorne, who have constructed Galois representations associated to not necessarily self-dual automorphic representations. The proof heavily relies on p-adic properties of automorphic representations.

The aim of the session “counting rational points on algebraic varieties” was to report on recent works on the existence, frequency and distribution of rational points on algebraic varieties. Thus the main themes were local to global principles, Manin's conjecture, developments of the Hardy-Littlewood method and the determinant method.